Develop a linear programming model which allows Northern Hi-Tec E. Ltd to determine how many computer peripheral devices should be produced to maximize profits assuming that all peripheral devices can be sold. The model formulations follow as below:

Definition of the decision variables

IM- Number of Internal Modem

EM- Number of External Modem

CB- Number of Circuit Board

CD- Number of CD Drive

HDD- Number of Hard Disk Drive

MB- Number of Memory Band

Identification and formulation of the objective function

Aim of Northern Hi-Tec Electronics Limited is to maximise profits.

Max Profit: 156.82IM+156.10EM+250.867CB+167.90CD+290.783HDD+274.2167MB

Identification and formulation of the constraints

Subjected to constraints:

Test Device 1- 2IM+3EM+5CB+6CD+4HDD+8MB?150

Test Device 2- IM+5EM+3CB+2CD+5HDD+5MB?130

Test Device 3- 5IM+2EM+3CB+3CD+3HDD+2MB?110

Test Device 4- 3IM+4EM+2CB+1CD+2HDD+3MB?102

Non-Negativity Constraints:

TD1?0, TD2?0, TD3?0, TD4?0

Solve the model you developed in part (b) using excel’s solver. Clearly state the value of the objective function and the optimal solution in the context of a problem.

The value of the objective function is:

Final value- Blank

Number of device IM – 157*168= 26376

Number of device EM – 156*0= 0

Number of device CB – 251*984= 246984

Number of device CD – 168*0= 0

Number of device HDD – 291*936= 272376

Number of device MB – 274*0= 0

26376+0+246984+0+272376+0= 545736

The optimal solution:

Number of device IM – 168

Number of device EM – 0

Number of device CB – 984

Number of device CD – 0

Number of device HDD – 936

Number of device MB – 0

Identify which of the constraint(s) are binding and which are not binding. How much slack (surplus) do the not-binding constraint(s) have? Be specific and give your answer in the context of the problem.

The constraints which are binding are: Test Device1, Test Device2 and Test Device3 with all slack of ‘0’. The only constraint which is not-binding is Test Device4 of a slack of ‘1776’.

If a reduction of £10 was made to the profit margin of ‘Internal Modem’. How would the optimal solution and the value of the objective function change? Be specific and give your answer in the context of the problem.

Looking at the sensitivity report the optimal solution will not change as Northern Hi-Tech Electronics Ltd can change the profit by either increasing or decreasing and by this it will make no affect on the optimal solution.

Also if a reduction of £10 was made to the profit margin of ‘Internal Modem’ it will make no change to the objective solution as looking at the sensitivity report we can see that the internal modem allows a decrease of up to £63.40 therefore it will have no affect on the objective solution. However looking at the objective function there will be an affect because each ‘Internal Modem’ will not cost £10 less so what is made from the 168 ‘Internal Modem’ it will be £10 less which means that the final value will decrease by 1680 of the objective function.

What is the value of an additional minute of time per week on test device 1? Test device 2? Test device 3? And test device4? Should Northern Hi-Tec E. Ltd. add more test device time? If so, on which test device?

The value of an additional minute per week for test device 1 is 21.77. The value for an additional minute of time per week for test device 2 is 30.85. The value of an additional minute per week on test device 3 is 16.48 and lastly the value of an additional minute of time per week on test device 4 is 0. All the values of an additional minute per week are made known in the sensitivity report and are shown in the shadow price.

Northern Hi-Tec E. Ltd. should add more test device time and that should be on test device 2. This is because for each additional minute it adds an extra 30.85 contribution. This is also the highest shadow price out of all the devices and also looking more in to the sensitivity report we can see that test device 2 has the most allowable additional minutes.

Write a short business report (of 1000 words) to advice Northern Hi-Tec E. Ltd. for its management decision. You will discuss the decision variables, constraints, optimal solutions, value of the objective function, slack variables, evaluate the impact of changes on variable labour costs (as in e) and the production level and combination of the peripheral devices that help Northern Hi-Tec E. Ltd maximize its profits.

My advice to Northern Hi-Tec E. Ltd is looking at the decision variables which are, Number of Internal Modem (IM), Number of External Modem (EM), Number of Circuit Board (CB), Number of CD Drive (CD), Number of Hard Disk Drive (HDD) and Number of Memory Band (MB). Their aim is to maximise profit from each decision variable listed above. So looking at the constraints for Northern Hi-Tec E. Ltd it is that Test Device1? 150, Test Device2? 130, Test Device3? 110 and Test Deivce4? 102 for all the decision variables. The value of the objective function is in total 545736 and this was worked out by using the sensitivity report and multiplying together the final value and the objective coefficient from the variable cells. After getting those values you add them together giving you the value of the objective function. The optimal solution is simply the final value from the sensitivity report under variable cells. The highest optimal solution if for Number of Circuit Board 984 followed by Number of Hard disk Drive 936 and then Number of Internal Modem of 168 the rest of the variables are 0.

The slack variables as we can see from the answer report Test Device 1, Test Device 2 and Test Device 3 are all binding but Test Device 4 is Non-binding because it had only used 4344 resources and had a total of 6120 to use therefore leaving 1776 still available.

Task 2

Write the mathematical expression of the linear programming problem for the objective function and state the constraints with the minimum required daily intake.

The mathematical expression for the objective function is 0.85BR+1.25CH+0.75CO+1.15PC.

The constraints with the minimum required daily intake are:

Daily calorie intake – 550

Daily chocolate intake – 165

Daily sugar intake – 280

Daily fat intake – 224

What is the optimal solution, in terms of the client’s daily intake, that satisfies the daily nutritional requirements at a minimum cost? Be specific and give your answer in the context of the problem.

The optimal solution in terms of the client’s daily intake that satisfies the daily nutritional requirements at a minimum cost is:

Number of brownies eaten daily BR – 2

Number of chocolate ice cream eaten daily CH – 0

Number of cola drank daily CO – 1

Number of pineapple cheesecake eaten daily PC – 1

Calculate the value of the objective function using the information provided in Table 3 and Table 4? Show your calculations and give your answer in the context of the problem.

The value of the objective function using table 3 and table 4.

Final Value- Blank

Number of brownies eaten daily BR – 2 – 0.85 x 2= 1.7

Number of chocolate ice cream eaten daily CH – 0 – 1.25 x 0= 0

Number of cola drank daily CO – 1 – 0.75 x 1= 0.75

Number of pineapple cheesecake eaten daily PC – 1 – 1.15 x 1= 1.15

1.7+0+0.75+1.15= 3.6

Are all resources fully utilised? Identify the resource(s), if any, that was not fully utilised or surplus. Be specific and give your explanation in the context of a problem.

Available amount of daily calorie intake 550

Amount used for daily calorie intake -1096.19= -546.19 material have been over used

Available amount of daily chocolate intake 165.00

Amount used of daily chocolate intake -165.00= All material have been used

Available amount of daily sugar intake 280.00

Amount used for daily sugar intake -280.00= All material have been used

Available amount of daily fat intake 224.00

Amount used for daily fat intake- -224.00= All material have been used

For any resource identified in part (D), what amount of this resource initially set as a minimum requirement? What amount of this resource has slack/surplus? Show your calculations and give your answer in the context of the problem.

All resources are fully utilised however daily calorie intake has been over used by 546.19 materials.

What is the impact on the optimal solution and the value of the objective function if the unit price for pineapple cheese-cake increases to £1.50? Give your answer in the context of the problem.

The impact on the optimal solution and the value of the objective function will remain the same as we have to increase the objective coefficient to £1.50 and we have an allowable increase of 0.55. As the objective coefficient is 1.15 we are in the region of the allowable increase.

What is a shadow price? The shadow price for daily calorie intake is 0. Explain the significance of this value in the context of the problem.

Shadow price is the change in the objective function for one additional unit of resource. The optimal value is of a problem that is acquired by relaxing the constraint by one unit and also delivers dominant visions in to problems to the decision maker. In other words shadow price is when a company is willing to give extra price for a unit of given resource.

Shadow price for daily calorie intake is ‘0’ because there is left over resources. Resources have not been fully utilised. No impact on profit. If one additional unit of this resources is obtained.

Write a short business report (of 600 words) to advice the consultant dietician that satisfies the client’s daily nutritional requirement at a minimum cost. You will discuss the decision variables, optimal solution, value of objective function, shadow price, the slack/surplus variables and evaluate the impact of changes on the combination and amounts of food types that provide the required nutrition at the least total cost.

My advice to the consultant dietician that satisfies the client’s daily nutritional requirement at a minimum cost is that looking at the decision variables and taking in to account the optimal solution for each clients daily intake is Brownies(BR), Pineapple Cheesecake(PC), Cola Drank(CO) and Chocolate Ice-cream(CH). We can see that the daily intake for brownies is 2, for pineapple cheesecake and cola drank is 1 and 0 daily intake of chocolate ice-cream. From looking at this we can see the brownies are being eaten the most daily, followed by pineapple cheesecake and cola drank on the same amount and then lastly chocolate ice-cream being eaten the least.

The value for the objective function is as follows Brownies (BR) – 1.7, Chocolate Ice-cream (CH) – 0, Cola Drank (C0) – 0.75 and Pineapple Cheesecake (PC) – 1.15. from this we can see that the value of the brownies are more meaning clients are eating more of them and seeing that the pineapple cheesecake and cola drank has the same optimal solution the value of the pineapple cheesecake is higher than the cola drank by clients and as for chocolate ice-cream it has 0 value.

My advice also to the consultant dietician on shadow pricing is that as we can see for daily calorie intake the shadow price is ‘0’ because not all resources have been fully utilised this made no impact on the profit if one additional unit of resource is obtained.